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--- Part Two ---
For some reason, your simulated results don't match what the experimental energy source engineers
expected. Apparently, the pocket dimension actually has [1m[37mfour spatial dimensions[0m, not
three.
The pocket dimension contains an infinite 4-dimensional grid. At every integer 4-dimensional
coordinate (x,y,z,w), there exists a single cube (really, a [1m[37mhypercube[0m) which is still
either [1m[37mactive[0m or [1m[37minactive[0m.
Each cube only ever considers its [1m[37mneighbors[0m: any of the 80 other cubes where any of
their coordinates differ by at most 1. For example, given the cube at x=1,y=2,z=3,w=4, its neighbors
include the cube at x=2,y=2,z=3,w=3, the cube at x=0,y=2,z=3,w=4, and so on.
The initial state of the pocket dimension still consists of a small flat region of cubes.
Furthermore, the same rules for cycle updating still apply: during each cycle, consider the
[1m[37mnumber of active neighbors[0m of each cube.
For example, consider the same initial state as in the example above. Even though the pocket
dimension is 4-dimensional, this initial state represents a small 2-dimensional slice of it. (In
particular, this initial state defines a 3x3x1x1 region of the 4-dimensional space.)
Simulating a few cycles from this initial state produces the following configurations, where the
result of each cycle is shown layer-by-layer at each given z and w coordinate:
Before any cycles:
z=0, w=0
.#.
..#
###
After 1 cycle:
z=-1, w=-1
#..
..#
.#.
z=0, w=-1
#..
..#
.#.
z=1, w=-1
#..
..#
.#.
z=-1, w=0
#..
..#
.#.
z=0, w=0
#.#
.##
.#.
z=1, w=0
#..
..#
.#.
z=-1, w=1
#..
..#
.#.
z=0, w=1
#..
..#
.#.
z=1, w=1
#..
..#
.#.
After 2 cycles:
z=-2, w=-2
.....
.....
..#..
.....
.....
z=-1, w=-2
.....
.....
.....
.....
.....
z=0, w=-2
###..
##.##
#...#
.#..#
.###.
z=1, w=-2
.....
.....
.....
.....
.....
z=2, w=-2
.....
.....
..#..
.....
.....
z=-2, w=-1
.....
.....
.....
.....
.....
z=-1, w=-1
.....
.....
.....
.....
.....
z=0, w=-1
.....
.....
.....
.....
.....
z=1, w=-1
.....
.....
.....
.....
.....
z=2, w=-1
.....
.....
.....
.....
.....
z=-2, w=0
###..
##.##
#...#
.#..#
.###.
z=-1, w=0
.....
.....
.....
.....
.....
z=0, w=0
.....
.....
.....
.....
.....
z=1, w=0
.....
.....
.....
.....
.....
z=2, w=0
###..
##.##
#...#
.#..#
.###.
z=-2, w=1
.....
.....
.....
.....
.....
z=-1, w=1
.....
.....
.....
.....
.....
z=0, w=1
.....
.....
.....
.....
.....
z=1, w=1
.....
.....
.....
.....
.....
z=2, w=1
.....
.....
.....
.....
.....
z=-2, w=2
.....
.....
..#..
.....
.....
z=-1, w=2
.....
.....
.....
.....
.....
z=0, w=2
###..
##.##
#...#
.#..#
.###.
z=1, w=2
.....
.....
.....
.....
.....
z=2, w=2
.....
.....
..#..
.....
.....
After the full six-cycle boot process completes, [1m[37m848[0m cubes are left in the
[1m[37mactive[0m state.
Starting with your given initial configuration, simulate six cycles in a 4-dimensional space.
[1m[37mHow many cubes are left in the active state after the sixth cycle?[0m
|
